Observables: photoelectron flux in the LF and MF

3.6. Observables: photoelectron flux in the LF and MF#

The observables of interest herein - the photoelectron flux as a function of energy, ejection angle, and time (see Fig. 1.1) - can be written quite generally as expansions in radial and angular basis functions. Various types and definitions are given in this section, including worked numerical examples.

3.6.1. Spherical harmonics#

The photoelectron flux as a function of energy, ejection angle, and time, can be written generally as an expansion in spherical harmonics:

(3.37)#\[ \begin{align} \bar{I}(\epsilon,t,\theta,\phi)=\sum_{L=0}^{2n}\sum_{M=-L}^{L}\bar{\beta}_{L,M}(\epsilon,t)Y_{L,M}(\theta,\phi) \end{align} \]

Here the flux in the laboratory frame (LF) or aligned frame (AF) is denoted $\bar{I}(\epsilon,t,\theta,\phi)$, with the bar signifying ensemble averaging, and the molecular frame flux by $I(\epsilon,t,\theta,\phi)$. Similarly, the expansion parameters $\bar{\beta}_{L,M}(\epsilon,t)$ include a bar for the LF/AF case. These observables are generally termed photoelectron angular distributions (PADs), often with a prefix denoting the reference frame, e.g. LFPADs, MFPADs, and the associated expansion parameters $\bar{\beta}_{L,M}(\epsilon,t)$ are generically termed anisotropy paramters. The polar coordinate system $(\theta,\phi)$ is referenced to an experimentally-defined axis in the LF/AF case (usually defined by the laser polarization), and the molecular symmetry axis in the MF, as illustrated in Fig. 1.1. Some arbitrary examples are given in Fig. 3.14, which illustrates both a range of distributions of increasing complexity, and some basic code to set $\beta_{L,M}$ parameters and visualise them; the values used as tabulated in Fig. 3.15.

Numerically, there are some choices and conventions which apply to the spherical harmonics. As noted in Sect. 2.3: “spherical harmonics are defined with the usual physics conventions: orthonormalised, and including the Condon-Shortley phase. Numerically they are implemented directly or via SciPy’s sph_harm function (see the SciPy docs for details [50].” For further details, including conversion routines, see the pySHtools [51, 52, 53, 54] documentation, and numerical examples below.

# Plot some distributions from specified BLMs

# Set specific LM coeffs by list with setBLMs, items are [l,m,value(s)]
# Multiple values are automatically assigned to an index 't'
from epsproc.sphCalc import setBLMs

BLM = setBLMs([[0,0,1,1,1,1,1,1],
               [1,0,0,0.5,0.8,1,0.5,0],[1,-1,0,0.5,0.8,1,0.5,0],[1,1,0,0.5,-0.5,1,0.5,0],
               [2,0,1,0.5,0,0,0.5,1],
               [4,2,0,0,0,0.5,0.8,1],[4,-2,0,0,0,0,-0.8,1]])

# Output a quick tabulation of the values with Pandas
BLM.to_pandas()

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	t	0	1	2	3	4	5
l	m
0	0	1.0	1.0	1.0	1.0	1.0	1.0
1	0	0.0	0.5	0.8	1.0	0.5	0.0
	-1	0.0	0.5	0.8	1.0	0.5	0.0
	1	0.0	0.5	-0.5	1.0	0.5	0.0
2	0	1.0	0.5	0.0	0.0	0.5	1.0
4	2	0.0	0.0	0.0	0.5	0.8	1.0
4	-2	0.0	0.0	0.0	0.0	-0.8	1.0

# Show harmonics info only
BLM.attrs['harmonics']

{'dtype': 'Complex harmonics',
 'kind': 'complex',
 'normType': 'ortho',
 'csPhase': True}

# Plot some PADs from BLMs
# Set the backend to 'pl' for an interactive surface plot with Plotly

# Explict row,column layout setting for figure
rc = [2,3]  

# Compute expansions from BLM parameters and return figure objext
dataPlot, figObj = ep.sphFromBLMPlot(BLM, facetDim='t', backend = plotBackend, rc=rc, plotFlag=True);

Fig. 3.14 Examples of angular distributions (expansions in spherical harmonics $Y_{L,M}$), for a range of cases indexed by $t$. Note that up-down asymmetry is associated with odd-$l$ contributions (e.g. $t=1,2$), breaking of cylindrical symmetry with $m\neq0$ terms (all $t>0$), and asymmetries in the (x,y) plane (skew/directionality) with different $\pm m$ terms (magnitude or phase, e.g. $t=2,3,4$). Higher-order $L,M$ terms have more nodes, and lead to more complex angular structures, as shown in the lower row ($t=3,4,5$).#

	t	0	1	2	3
l	m
0	0	1.0	1.0	1.0	1.0
1	0	0.0	0.5	0.8	1.0
2	0	1.0	0.5	0.0	0.0
4	2	0.0	0.0	0.0	0.5
4	-2	0.0	0.0	0.0	0.5

Fig. 3.15 Values used for the plots in Fig. 3.14.#

In general, the spherical harmonic rank and order $(L,M)$ of Eq. (3.37) are constrained by experimental factors in the LF or AF, and $n$ is effectively limited by the molecular alignment (which is correlated with the photon-order for gas phase experiments, or conservation of angular momentum in the LF more generally [107]), but in the MF is defined by the maximum continuum angular momentum $n=l_{max}$ imparted by the scattering event [108] (note lower-case $l$ here refers specifically to the continuum photoelectron wavefunction, see Eq. (3.5)).

For basic cases these limits may be low: for instance, a simple 1-photon photoionization event ($n=1$) from an isotropic ensemble (zero net ensemble angular momentum) defines $L_{max}=2$; for cylindrically symmetric cases (i.e. $D_{\infty h}$ symmetry) $M=0$ only. For MF cases, $l_{max}=4$ is often given as a reasonable rule-of-thumb for the continuum - hence $L_{max}=8$ - although in practice higher-$l$ may be populated. Some realistic example cases are discussed later (Part II), see also ref. [2] for more discussion and complex examples.

In general, these observables may also be dependent on various other parameters; in Eq. (3.37) two such parameters, $(\epsilon,t)$, are included, as the usual variables of interest. Usually $\epsilon$ denotes the photoelectron energy, and $t$ is used in the case of time-dependent (usually pump-probe) measurements. As discussed in Sect. 3.1, the origin of such dependencies may be complicated but, in general, the associated photoionization matrix elements are energy-dependent, and time-dependence may also appear for a number of intrinsic or extrinsic (experimental) reasons, e.g. electronic or nuclear dynamics, rotational (alignment) dynamics, electric field dynamics etc. In many cases only one particular aspect may be of interest, so $t$ can be used as a generic label to index changes as per Fig. 3.14.

3.6.2. Symmetrized harmonics#

Symmetrized (or generalised) harmonics, which essentially provide correctly symmetrized expansions of spherical harmonics ($Y_{LM}$) functions for a given irreducible representation, $\Gamma$, of the molecular point-group can be defined by linear combinations of spherical harmonics [109, 110, 111]:

(3.38)#\[ X_{hl}^{\Gamma\mu*}(\theta,\phi)=\sum_{\lambda}b_{hl\lambda}^{\Gamma\mu}Y_{l,\lambda}(\theta,\phi) \]

where:

$\Gamma$ is an irreducible representation;
$(l, \lambda)$ define the usual spherical harmonic indices (rank, order), but note the use of $(l, \lambda)$ by convention, since these harmonics are usually referenced to the MF;
$b_{hl\lambda}^{\Gamma\mu}$ are symmetrization coefficients;
index $\mu$ allows for indexing of degenerate components (note here the unfortunate convention that the label $\mu$ is also used for photon projection terms in general, as per Sect. 3.3.2 - in ambiguous cases the symmetrization term will instead be labelled herein as $\mu_X$, although in many cases may actually be redundant and safely dropped from the symmetrization coefficients);
$h$ indexes cases where multiple components are required with all other quantum numbers identical.

Analogously to Eq. (3.37), a general expansion of an observable in the symmetrized harmonic basis set can then be defined as:

(3.39)#\[ \bar{I}^{\Gamma}(\epsilon,t,\theta,\phi) = \sum_{\Gamma\mu hl}\bar{\beta}_{hl}^{\Gamma\mu}(\epsilon,t)X_{hl}^{\Gamma\mu*}(\theta,\phi) \]

Alternatively, by substitution into Eq. (3.37), and assigning $l=L$ and $\lambda=M$, a general symmetrized expansion may also be defined as:

(3.40)#\[ \begin{align} \bar{I}(\epsilon,t,\theta,\phi)=\sum_{\Gamma\mu h}\sum_{L=0}^{2n}\sum_{M=-L}^{L}b_{hLM}^{\Gamma\mu}\bar{\beta}_{L,M}(\epsilon,t)Y_{L,M}(\theta,\phi) \end{align} \]

However, in many cases the symmetrization coefficients are subsumed into the $\beta_{L,M}$ terms (or underlying matrix elements); in this case a simplified symmetrized expansion can be defined as:

(3.41)#\[ \begin{align} \bar{I}^{\Gamma}(\epsilon,t,\theta,\phi)=\sum_{L=0}^{2n}\sum_{M=-L}^{L}\bar{\beta}^{\Gamma}_{L,M}(\epsilon,t)Y_{L,M}(\theta,\phi) \end{align} \]

Where the expansion is defined for a given symmetry and irreducible representation with the shorthand $\Gamma$; in many systems a single label may be sufficient here, since allowed $(L,M)$ terms will be defined uniquely by irreducible representation, although multiple quantum numbers may be required for unique definition in the most general cases as per Eq. (3.38) (e.g. for cases with degenerate components). Further details and usage in relation to channel functions are also discussed in Sect. 3.3 (see, in particular, Eq. (3.13) for a similar general case), and in relation to fitting for specific cases in Part II.

The exact form of these coefficients will depend on the point-group of the system, see, e.g. Refs. [111, 112]. Numerical routines for the generation of symmetrized harmonics are implemented in Photoelectron Metrology Toolkit [3]: point-groups, character table generation and symmetrization (computing $b_{hl\lambda}^{\Gamma\mu}$ parameters) is handled by libmsym [55, 56]; additional handling also makes use of pySHtools [51, 52, 53, 54].

A brief numerical example is given below, and more details can be found in the PEMtk documentation [14]. In this case, full tabulations of the parameters list all $b_{hLM}^{\Gamma\mu}$ for each irreducible representation, and the corresponding PADs are illustrated in Fig. 3.17.

# Import class
from pemtk.sym.symHarm import symHarm

# Compute hamronics for Td, lmax=4
sym = 'Td'
lmax=6

symObj = symHarm(sym,lmax)

# Character tables can be displayed - this will render directly in a notebook.
symObj.printCharacterTable()

Show code cell output Hide code cell output

*** Mapping coeffs to ePSproc dataType = matE
Remapped dims: {'C': 'Cont', 'mu': 'it'}
Added dim Eke
Added dim Targ
Added dim Total
Added dim mu
Added dim Type
Found dipole symmetries: 
{'T2': {'m': [-1, 1, 0, -1, 1], 'pol': ['x', 'z', 'y']}}

/home/jovyan/github/PEMtk/pemtk/sym/_dipoleTerms.py:102: FutureWarning:

iteritems is deprecated and will be removed in a future version. Use .items instead.

		E	C2^1	S4^1	σd	C3^1
Character	dim
A1	1	1.0	1.0	1.0	1.0	1.0
A2	1	1.0	1.0	-1.0	-1.0	1.0
E	2	2.0	2.0	0.0	0.0	-1.0
T1	3	3.0	-1.0	1.0	-1.0	0.0
T2	3	3.0	-1.0	-1.0	1.0	0.0

		E	C2^1	S4^1	σd	C3^1
Character	dim
A1	1	1.0	1.0	1.0	1.0	1.0
A2	1	1.0	1.0	-1.0	-1.0	1.0
E	2	2.0	2.0	0.0	0.0	-1.0
T1	3	3.0	-1.0	1.0	-1.0	0.0
T2	3	3.0	-1.0	-1.0	1.0	0.0

Fig. 3.16 Example character table for Td symmetry generated with the Photoelectron Metrology Toolkit [3] wrapper for libmsym [55, 56].#

# The full set of expansion parameters can be tabulated

# pd.set_option('display.max_rows', 100)

symObj.displayXlm()  # Display values (note this defaults to REAL harmonics)
# symObj.displayXlm(YlmType='comp')   # Display values for COMPLEX harmonic expansion.

Show code cell output Hide code cell output

				b
			l	0	1	2	3	4	5	6
Character ($\Gamma$)	SALC (h)	PFIX ($\mu$)	m
A1	0	0	0	1.0
	1	0	-2				1.0
	2	0	0					0.764
	2	0	4					0.645
	3	0	0							0.354
	3	0	4							-0.935
A2	0	0	2							0.829
A2	0	0	6							-0.559
E	0	0	0			0.5
		0	2			0.866
		1	0			-0.866
		1	2			0.5
	1	0	0					0.323
			2					-0.866
			4					-0.382
		1	0					-0.559
			2					-0.5
			4					0.661
	2	0	-4						0.866
		0	-2						0.5
		1	-4						0.5
		1	-2						-0.866
	3	0	0							0.468
			2							0.484
			4							0.177
			6							0.718
		1	0							-0.81
			2							0.28
			4							-0.306
			6							0.415
T1	0	0	1				0.791
		0	3				0.612
		1	2				-1.0
		2	-3				-0.612
		2	-1				0.791
	1	0	-3					0.354
		0	-1					0.935
		1	-4					-1.0
		2	1					0.935
		2	3					-0.354
	2	0	1						0.661
			3						-0.306
			5						-0.685
		1	2						1.0
		2	-5						-0.685
			-3						0.306
			-1						0.661
	3	0	-5							0.586
			-3							0.685
			-1							-0.433
		1	-4							-1.0
		2	1							-0.433
			3							-0.685
			5							0.586
T2	0	0	1		1.0
		1	0		-1.0
		2	-1		1.0
	1	0	-1			1.0
		1	-2			-1.0
		2	1			1.0
	2	0	1				0.612
		0	3				-0.791
		1	0				1.0
		2	-3				0.791
		2	-1				0.612
	3	0	-3					0.935
		0	-1					-0.354
		1	-2					-1.0
		2	1					-0.354
		2	3					-0.935
	4	0	1						0.75
			3						0.27
			5						0.604
		1	0						-0.645
		1	4						-0.764
		2	-5						0.604
			-3						-0.27
			-1						0.75
	5	0	3						0.913
		0	5						-0.408
		1	0						0.764
		1	4						-0.645
		2	-5						-0.408
		2	-3						-0.913
	6	0	-5							0.81
			-3							-0.496
			-1							0.313
		1	-6							-0.134
		1	-2							-0.991
		2	1							0.313
			3							0.496
			5							0.81
	7	0	-3							0.535
		0	-1							0.845
		1	-6							-0.991
		1	-2							0.134
		2	1							0.845
		2	3							-0.535

# To plot using ePSproc/PEMtk class, these values can be converted to ePSproc BLM data type...

# Run conversion - the default is to set the coeffs to the 'BLM' data type
symObj.toePSproc()

# Set to new key in data class
data.data['symHarm'] = {}

for dataType in ['BLM']:  #['matE','BLM']:
    # Select expansion in complex harmonics
    data.data['symHarm'][dataType] = symObj.coeffs[dataType]['b (comp)']  
    data.data['symHarm'][dataType].attrs = symObj.coeffs[dataType].attrs
    
# Plot full harmonics expansions, plots by symmetry
# Note 'squeeze=True' to force drop of singleton dims may be required.
# data.padPlot(keys='symHarm',dataType='BLM', facetDims = ['Cont'], squeeze = True, backend=plotBackend)

# As above, with some additional layout options
rc = [2,3]  # Explict layout setting
data.padPlot(keys='symHarm',dataType='BLM', facetDims = ['Cont'], squeeze = True, backend=plotBackend, 
             rc = rc, plotFlag=False, returnFlag=True)
figObj = data.data['symHarm']['plots']['BLM']['polar'][0]

# And GLUE for display later with caption
gluePlotly("symHarmPADs", figObj)

Fig. 3.17 Examples of angular distributions from expansions in symmetrized harmonics $X_{hl}^{\Gamma\mu*}(\theta,\phi)$, for all irreducible representations in Td symmetry ($l_{max}=$6). (Note $A_2$ only has components for $l\geq 6$.)#

3.6.2.1. Real & complex forms#

By convention, the complex form of the spherical harmonics are usually used for photoionization problems. However, real harmonics are also in common use (and have already appeared in the numerical routines above). The relationships can be defined as (per the Wikipaedia definitions [113]):

(3.42)#\[\begin{split} \begin{aligned} Y_{\ell m}&={\begin{cases}{\dfrac {i}{\sqrt {2}}}\left(Y_{\ell }^{m}-(-1)^{m}\,Y_{\ell }^{-m}\right)&{\text{if}}\ m\lt0 \\Y_{\ell }^{0}&{\text{if}}\ m=0 \\{\dfrac {1}{\sqrt {2}}}\left(Y_{\ell }^{-m}+(-1)^{m}\,Y_{\ell }^{m}\right)&{\text{if}}\ m\gt0.\end{cases}} \\&={\begin{cases}{\dfrac {i}{\sqrt {2}}}\left(Y_{\ell }^{-|m|}-(-1)^{m}\,Y_{\ell }^{|m|}\right)&{\text{if}}\ m\lt0 \\Y_{\ell }^{0}&{\text{if}}\ m=0 \\{\dfrac {1}{\sqrt {2}}}\left(Y_{\ell }^{-|m|}+(-1)^{m}\,Y_{\ell }^{|m|}\right)&{\text{if}}\ m\gt0.\end{cases}} \\&={\begin{cases}{\sqrt {2}}\,(-1)^{m}\,\Im [{Y_{\ell }^{|m|}}]&{\text{if}}\ m\lt0 \\Y_{\ell }^{0}&{\text{if}}\ m=0 \\{\sqrt {2}}\,(-1)^{m}\,\Re [{Y_{\ell }^{m}}]&{\text{if}}\ m\gt0.\end{cases}} \end{aligned} \end{split}\]

Where the notation here uses $Y_{\ell m}$ for real harmonics, and $Y_{\ell }^{m}$ for complex.

Conversion between types is handled in the ePSproc codebase [24, 25, 26] either directly or via the pySHtools [51, 52, 53, 54] library, and objects usually have the type specified in their metadata (if missing, they are assumed to be complex). The symmetry routines outlined above automatically compute both types, and these are available in the output data structure (see the PEMtk documentation [14] for further details and examples, and the “Working with real spherical harmonics” note from the ePSproc documentation [26], and the relevant pySHtools documentation pages).

# Display complex values
symObj.displayXlm(YlmType='comp')   # Display values for COMPLEX harmonic expansion.

Show code cell output Hide code cell output

				b
			l	0	1	2	3	4	5	6
Character ($\Gamma$)	SALC (h)	PFIX ($\mu$)	m
A1	0	0	0	(1+0j)
	1	0	-2				(-0.7071067811865475+0j)
	1	0	2				(0.7071067811865475+0j)
	2	0	-4					(0.45643546458763845+0j)
			0					(0.7637626158259733+0j)
			4					(0.45643546458763845+0j)
	3	0	-4							(-0.6614378277661477-0j)
			0							(0.35355339059327356+0j)
			4							(-0.6614378277661477+0j)
A2	0	0	-6							(-0.39528470752104755-0j)
			-2							(0.5863019699779286+0j)
			2							(0.5863019699779286+0j)
			6							(-0.39528470752104755+0j)
E	0	0	-2			(0.35355339059327384+0j)
			0			(-0.8660254037844385+0j)
			2			(0.35355339059327384+0j)
		1	-2			(0.35355339059327384+0j)
			0			(-0.8660254037844385+0j)
			2			(0.35355339059327384+0j)
	1	0	-4					(0.4677071733467426+0j)
			-2					(-0.3535533905932736-0j)
			0					(-0.5590169943749477+0j)
			2					(-0.3535533905932736+0j)
			4					(0.4677071733467426+0j)
		1	-4					(0.4677071733467426+0j)
			-2					(-0.3535533905932736-0j)
			0					(-0.5590169943749477+0j)
			2					(-0.3535533905932736+0j)
			4					(0.4677071733467426+0j)
	2	0	-4						(-0.3535533905932735+0j)
			-2						(0.6123724356957947+0j)
			2						(-0.6123724356957947+0j)
			4						(0.3535533905932735+0j)
		1	-4						(-0.3535533905932735+0j)
			-2						(0.6123724356957947+0j)
			2						(-0.6123724356957947+0j)
			4						(0.3535533905932735+0j)
	3	0	-6							(0.293150984988964+0j)
			-4							(-0.2165063509461096-0j)
			-2							(0.19764235376052366+0j)
			0							(-0.8100925873009829+0j)
			2							(0.19764235376052366+0j)
			4							(-0.2165063509461096+0j)
			6							(0.293150984988964+0j)
		1	-6							(0.293150984988964+0j)
			-4							(-0.2165063509461096-0j)
			-2							(0.19764235376052366+0j)
			0							(-0.8100925873009829+0j)
			2							(0.19764235376052366+0j)
			4							(-0.2165063509461096+0j)
			6							(0.293150984988964+0j)
T1	0	0	-3				(-0.4330127018922194-0j)
			-1				(0.5590169943749473+0j)
			1				(-0.5590169943749473-0j)
			3				(0.4330127018922194-0j)
		1	-2				(-0.7071067811865475-0j)
		1	2				(-0.7071067811865475+0j)
		2	-3				(0.4330127018922194+0j)
			-1				(-0.5590169943749473+0j)
			1				(-0.5590169943749473-0j)
			3				(0.4330127018922194-0j)
	1	0	-3					(0.2500000000000001+0j)
			-1					(-0.6614378277661476+0j)
			1					(-0.6614378277661476-0j)
			3					(0.2500000000000001-0j)
		1	-4					(0.7071067811865475+0j)
		1	4					(-0.7071067811865475+0j)
		2	-3					(-0.2500000000000001-0j)
			-1					(0.6614378277661476+0j)
			1					(-0.6614378277661476-0j)
			3					(0.2500000000000001-0j)
	2	0	-5						(-0.4841229182759272-0j)
			-3						(0.21650635094610968+0j)
			-1						(0.4677071733467425+0j)
			1						(-0.4677071733467425-0j)
			3						(-0.21650635094610968-0j)
			5						(0.4841229182759272-0j)
		1	-2						(0.7071067811865475+0j)
		1	2						(0.7071067811865475+0j)
		2	-5						(0.4841229182759272+0j)
			-3						(-0.21650635094610968+0j)
			-1						(-0.4677071733467425+0j)
			1						(-0.4677071733467425-0j)
			3						(-0.21650635094610968-0j)
			5						(0.4841229182759272-0j)
	3	0	-5							(-0.414578098794425+0j)
			-3							(0.4841229182759269+0j)
			-1							(0.30618621784789757+0j)
			1							(0.30618621784789757-0j)
			3							(0.4841229182759269-0j)
			5							(-0.414578098794425-0j)
		1	-4							(0.7071067811865475+0j)
		1	4							(-0.7071067811865475+0j)
		2	-5							(0.414578098794425+0j)
			-3							(-0.4841229182759269-0j)
			-1							(-0.30618621784789757-0j)
			1							(0.30618621784789757-0j)
			3							(0.4841229182759269-0j)
			5							(-0.414578098794425-0j)
T2	0	0	-1		(0.7071067811865475+0j)
		0	1		(-0.7071067811865475-0j)
		1	0		(-1+0j)
		2	-1		(-0.7071067811865475+0j)
		2	1		(-0.7071067811865475-0j)
	1	0	-1			(-0.7071067811865475+0j)
		0	1			(-0.7071067811865475-0j)
		1	-2			(0.7071067811865475+0j)
		1	2			(-0.7071067811865475+0j)
		2	-1			(0.7071067811865475+0j)
		2	1			(-0.7071067811865475-0j)
	2	0	-3				(0.5590169943749472+0j)
			-1				(0.43301270189221935+0j)
			1				(-0.43301270189221935-0j)
			3				(-0.5590169943749472-0j)
		1	0				(1+0j)
		2	-3				(-0.5590169943749472+0j)
			-1				(-0.43301270189221935+0j)
			1				(-0.43301270189221935-0j)
			3				(-0.5590169943749472-0j)
	3	0	-3					(0.6614378277661477+0j)
			-1					(0.25000000000000006+0j)
			1					(0.25000000000000006-0j)
			3					(0.6614378277661477-0j)
		1	-2					(0.7071067811865475+0j)
		1	2					(-0.7071067811865475+0j)
		2	-3					(-0.6614378277661477-0j)
			-1					(-0.25000000000000006-0j)
			1					(0.25000000000000006-0j)
			3					(0.6614378277661477-0j)
	4	0	-5						(-0.2886751345948129-0j)
			-3						(-0.6454972243679028-0j)
			-1						(0.5303300858899107+0j)
			1						(-0.5303300858899107-0j)
			3						(0.6454972243679028-0j)
			5						(0.2886751345948129-0j)
		1	-4						(-0.45643546458763823-0j)
			0						(0.7637626158259736+0j)
			4						(-0.45643546458763823+0j)
		2	-5						(0.2886751345948129+0j)
			-3						(0.6454972243679028+0j)
			-1						(-0.5303300858899107+0j)
			1						(-0.5303300858899107-0j)
			3						(0.6454972243679028-0j)
			5						(0.2886751345948129-0j)
	5	0	-5						(-0.2886751345948129-0j)
			-3						(-0.6454972243679028-0j)
			3						(0.6454972243679028-0j)
			5						(0.2886751345948129-0j)
		1	-4						(-0.45643546458763823-0j)
			0						(0.7637626158259736+0j)
			4						(-0.45643546458763823+0j)
		2	-5						(0.2886751345948129+0j)
			-3						(0.6454972243679028+0j)
			3						(0.6454972243679028-0j)
			5						(0.2886751345948129-0j)
	6	0	-5							(-0.57282196186948+0j)
			-3							(0.3779644730092276+0j)
			-1							(-0.5976143046671966+0j)
			1							(-0.5976143046671966-0j)
			3							(0.3779644730092276-0j)
			5							(-0.57282196186948-0j)
		1	-6							(0.700764888226735+0j)
			-2							(-0.09449111825230713+0j)
2									(0.09449111825230713+0j)
6									(-0.700764888226735+0j)
2		-5							(0.57282196186948+0j)
		-3							(-0.3779644730092276-0j)
		-1							(0.5976143046671966+0j)
		1							(-0.5976143046671966-0j)
		3							(0.3779644730092276-0j)
		5							(-0.57282196186948-0j)
7	0	-3							(0.3779644730092276+0j)
		-1							(-0.5976143046671966+0j)
		1							(-0.5976143046671966-0j)
		3							(0.3779644730092276-0j)
	1	-6							(0.700764888226735+0j)
		-2							(-0.09449111825230713+0j)
		2							(0.09449111825230713+0j)
		6							(-0.700764888226735+0j)
	2	-3							(-0.3779644730092276-0j)
		-1							(0.5976143046671966+0j)
		1							(-0.5976143046671966-0j)
		3							(0.3779644730092276-0j)

# Access complex values from SH tools objects
# SHtools object are stored in nested dicts by character and type
print(symObj.coeffs['SH']['A1'].keys())
print(symObj.coeffs['SH']['A1']['comp'])

dict_keys(['real', 'comp'])
kind = 'complex'
normalization = '4pi'
csphase = -1
lmax = 6
error_kind = None
header = None
header2 = None
name = None
units = None

3.6.3. Legendre polynomials#

Finally, it is of note that Legendre polynomial expansions are also in common use in the description of photoionization obsevables. These are suitable for cylindrically symmetric cases only, and form a subset of the general spherical harmonic case. Using the same notation as Eq. (3.37), the 1D expansion can be given as:

(3.43)#\[ \bar{I}(\epsilon,t,\theta)=\sum_{L=0}^{2n}\bar{\beta}_{L}(\epsilon,t)P_{L}(\cos(\theta)) \]

Where $P_{L}(\cos(\theta))$ are Legendre polynomials in $\cos(\theta)$ (equivalently, associated Legendre polynomials $P_{L}^{M=0}(\cos(\theta))$). Note that, since the normalisation is different, care must be taken when comparing associated anisotropy parameters between Legendre polynomial and spherical harmonic expansions. Specifically:

(3.44)#\[ \beta^{Sph}_{L,0} = \sqrt{(2L+1)/4\pi}\beta^{Lg}_{L} \]

Where $Sph$ and $Lg$ labels have been added to make explicit that the expansion parameters in the spherical harmonic and Legndre polynomial basis sets respectively.

Herein only spherical harmonic expansions are used, but the ePSproc codebase [24, 25, 26] does include a conversion routine to convert expansion parameters as required. Again more information can be found in the ePSproc documentation [26], particularly the “Working with spherical harmonics” notebook.

# Set example BLMs and convert to Lg basis
# Note 'renorm=True' setting to renorm by B0 (will affect abs value of B0, but not form of distribution)
# Note also 'harmonics' and 'normType' specifications in output data.

BLMsph = setBLMs([[0,0,1],[1,0,0.5],[2,0,0.8]]).squeeze(drop=True)
BLMlg = ep.util.conversion.conv_BL_BLM(BLMsph, to = 'lg', renorm = True)
BLMlg

	t	0	1	2	3
l	m
0	0	1.0	1.0	1.0	1.0
1	0	0.0	0.5	0.8	1.0
2	0	1.0	0.5	0.0	0.0
4	2	0.0	0.0	0.0	0.5
4	-2	0.0	0.0	0.0	0.5

	t	0	1	2	3
l	m
0	0	1.0	1.0	1.0	1.0
1	0	0.0	0.5	0.8	1.0
2	0	1.0	0.5	0.0	0.0
4	2	0.0	0.0	0.0	0.5
4	-2	0.0	0.0	0.0	0.5

Observables: photoelectron flux in the LF and MF

Contents

3.6. Observables: photoelectron flux in the LF and MF#

3.6.1. Spherical harmonics#

3.6.2. Symmetrized harmonics#

3.6.2.1. Real & complex forms#

3.6.3. Legendre polynomials#

	t	0	1	2	3
l	m
0	0	1.0	1.0	1.0	1.0
1	0	0.0	0.5	0.8	1.0
2	0	1.0	0.5	0.0	0.0
4	2	0.0	0.0	0.0	0.5
4	-2	0.0	0.0	0.0	0.5